Bend flat, circular object into half sphere
Use a Cast Modifier.

For some reason you will have to move all you circle vertex up in local Z axis for it to work for some reason, then have to eye-ball the value to reach an aproximate half sphere.

In my case, move the circle [2] units in local Z axis in edit mode, then add a Cast modifier set to Sphere. Adjust tge Factor value.

Alternatively use the Control Object option, set it to an empty object to control the deformation center and move it down in Z axis as necessary, if you don't want to move the circle object center.